Left derived functor

Definition

A left derived functor is a construction in homological algebra that extends a given functor defined on a category to the derived category, providing a way to measure the failure of exactness. It is created by applying a sequence of projective resolutions to an object and then applying the original functor to these resolutions. This process allows one to capture important topological and algebraic properties of the objects involved, revealing deeper connections in their structures.

5 Must Know Facts For Your Next Test

1. Left derived functors are denoted typically as $L_nF$, where $F$ is the original functor and $n$ indicates the nth derived functor in the sequence.
2. They are especially useful when dealing with additive or abelian categories, where one can construct projective resolutions to derive meaningful results.
3. The first left derived functor, $L_1F$, corresponds to the derived functor of an additive functor, which measures how far the functor is from being left exact.
4. Higher left derived functors provide deeper insights into the structure of the objects involved, capturing additional information such as cohomology groups.
5. In categorical terms, left derived functors allow for transferring properties from a category of modules to a derived category, establishing connections that are pivotal for further studies in homological algebra.

Review Questions

• How does the construction of left derived functors utilize projective resolutions?
  • The construction of left derived functors relies on taking projective resolutions of an object before applying the original functor. This means that for an object $X$, one finds a projective resolution $P \to X \to 0$ and then applies the functor $F$ to this sequence. The resulting sequence captures how well $F$ behaves with respect to exactness, providing valuable information about the object and its structure through derived functors.
• What role do left derived functors play in measuring exactness in algebraic contexts?
  • Left derived functors are essential for measuring the failure of exactness in algebraic contexts by extending a functor into the realm of derived categories. They identify how far a given functor is from being left exact by calculating cohomological dimensions. For example, if $L_1F(X)$ is nonzero, it indicates that $F$ does not preserve all short exact sequences involving $X$, thus highlighting potential shortcomings in its algebraic structure.
• Evaluate the significance of left derived functors in broader mathematical theories beyond homological algebra.
  • Left derived functors have far-reaching implications beyond homological algebra, particularly in areas like algebraic geometry and representation theory. They provide tools for understanding sheaf cohomology in algebraic geometry, linking geometric properties with algebraic structures. Moreover, they play a crucial role in understanding derived categories, allowing mathematicians to study complex relationships between various algebraic objects and their morphisms, thereby enriching overall mathematical discourse.

Related terms

Exact Sequence: An exact sequence is a sequence of algebraic objects and morphisms between them such that the image of one morphism equals the kernel of the next, reflecting a certain balance of structure.

Projective Resolution: A projective resolution of an object is an exact sequence of projective modules that approximates the object, allowing one to study its properties through homological methods.

Derived Category: The derived category is a construction that allows one to work with complexes of objects and morphisms, providing a framework for studying homological properties in a more flexible manner.